Win the 12th, 13th, and 14th waypoint races in New York in arcade mode.

Type-S car

Enter career mode. Defeat all the hookmen in New York, fly to London, then defeat all three hookmen there. You must defeat all three hookmen in both cities. When you finally defeat the world champ, the Type-S car will be unlocked in your arcade garage.

Bonus cars

Look for red dots that appear on various levels. Place your car on the dot until a click sound is heard to unlock a new car. Some of the dots can be found at the following locations:

London

Go near the center of the map. The dot is inside the Parliament building (block building with pillars on the outside). Drive through the window to get inside.
Go near the bottom of the map and find the building with a circle street around it. One of the buildings across is red, with the word "Angel" on it. Go behind it to find a type of ramp. Go up the ramp with a speed of about 60 mph, then go to the right. The red dot should be in front or behind you.
Find the reddish building with advertisements and an animated screen on it. Also, there is a box that almost looks like a padded cell in the middle of the street near it. Find the ramp leading up along the side of the building. Before you exit the ramp, you should be going approximately 30 to 45 mph (depending on the weight of your car). After hitting the roof immediately slam on the brakes. The dot should be to the right of you.
Once you start in London, make a 180 degree turn and head towards the blue blinking "Speed" sign. Go into the ramp at full speed. Go over on to the next roof and hit the brakes. Then, make a left turn and you will see the dot.

New York

Go near the park above the big tunnel. The dot is on top of a roof that you need to jump to reach.
Go near the aircraft carrier. The dot is inside the second story of a building that you need to jump to enter.